Entropy and reduced distance for Ricci expanders

The Journal of Geometric Analysis Volume 15, Number 1, 2005

Entropy and Reduced Distance

Expanders

for Ricci

By

Michael Feldman, Tom Ilmanen, and Lei Ni

ABSTRACT. Perelman has discovered ~'o integral quantities, the shrinker entropy ~'V and the (backward) reduced volume, that are monotone under the Ricci flow Ogij /Ot = - 2 R i j and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy ~A; + is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman. it follows from a differential inequality for a Harnack-like quanti~' for the conjugate heat equation, and leads to functionals #+ and v+. The forward reduced volume O+ is monotone in general and constant exactly on expanders.

A natural conjecture asserts that g ( t ) / t converges as t --+ ~ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts), l f the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantiO, that is constant on expanders; these include vol(g)/t n/2 (Hamilton) and ~. (Perelman), as well as our new quantities. In general, we show that, tfvol(g) grows like t n/2 (maximal volume growth) then W + , O+ and ~ remain bounded (in their appropriate ways)for all time. We attempt a sharp formulation of the conjecture.

Small, large and distant parts of a Ricci flow are known to be modeled by various kinds of Ricci solitons: Steady, shrinking, and expanding. Perelman has discovered monotone quantities

corresponding to the first two of these. The functional )r is monotone on the Ricci flow and constant precisely on steadies [25, Section 1]. The shrinking, or localizing, entropy W is monotone in general and constant precisely on shrinking solitons [25, Section 3]. Closely related is the notion of the (backward) reduced volume [25, Section 7]. The latter two show that developing singularities and ancient histories are modeled on shrinkers, when an appropriate blowup, or blowdown is taken [25, 27].

In this note, by tweaking some signs, we observe similar monotonicity formulae for the expanding case. In Section 1, analogous to [25, Section 3], we define an expanding, or delocalizing entropy W + , which is monotone in general and constant precisely on expanders. It follows from

Math Subject Classifications. Primary: 58G11.

Key Words and Phrases. Entropy, Ricci flow, Ricci expanders, asymptotic behavior, reduced volume, monotonicity, Li-Yau-Hamilton inequality.

Acknowledgements and Notes. First author was partially supported by NSF grant DMS-0200644; second author was partially supported by Schweizerische Nationalfonds grant 21-66743.01 ; third author was partially supported by NSF grants and an Alfred P. Sloan Fellowship.

9 2005 The Journal of Geometric Analysis ISSN 1050-6926

5 0 Michael Feldman, Tom llmanen, and Lei Ni

a differential inequality for the conjugate heat equation similar to [25, Section 9]; see also [20]. We also construct functionals # + ( g , ~r) and v+(g) from W+.

In Section 2, analogous to [25, Section 7], we present a forward reduced distance ~+ and its corresponding forward reduced volume, which is again monotone in general, and constant exactly on expanders.

Ourformulae may be compared to those for vol(g) / t n/2 (Hamilton) andofX(g) := vol(g) 2/n ~.(g) (Perelman). All of these monotonicities have implications for the convergence of the rescaled Ricci flow as t ~ ~ . If g(t) is a compact Ricci flow that exists for all t > 0, and the volume vol(g) grows like t n/2 (maximal volume growth), we observe in Proposition 1.4 that W+ remains bounded. A similar result holds for Perelman's scaled )~-functional, and for the forward reduced volume (Proposition 2.3). This suggests that (ignoring zero-volume collapsing parts) the long- term limit o f g ( t ) / t is a negative Einstein manifold, possibly with singularities; see Conjecture 1.6. Our results may be set against recently discovered entropy formulae and pointwise differential inequalities in other settings: For the standard heat equation on a manifold of non-negative Ricci curvature [22], and for the forward conjugate heat equation ut = Au § Ru on a two-dimensional Ricci flow [23] and (matrix Harnack!) on a Kahler-Ricci flow [24]. 1

It is also instructive to compare the Ricci flow to the mean curvature flow in Euclidean space. The monotonicity formulae for 5 t-, W, W+, and for the corresponding reduced volumes, are analogous to the following three monotone energies on a mean curvature flow Mt in RN:

d e-lX-xol2/41to -t) d-7 fMt (-4~r t)n/2 d f M e(X'X~176 < O, t C ~ , dt t _<0, t < O , d elx-rol2/4(t-to ) d"-[ fMt (43rt)n/2 < 0 , t > 0 .

The first is Huisken's well-known result [15] and is constant precisely on shrinkers. The second picks out steadies (translators) [16], whereas the third is stationary on expanders [19].

In the non-compact case, for both flows, the energy of a steady, or expander is typically infinite, so in this case the monotonicity formulae will not be useful in proving convergence to an asymptotic soliton. 2 This is unfortunate, because non-compact expanders are needed for modeling the relaxation out of a previously formed singularity of conic type; see, for example, [2, 18, 10] and remarks after Proposition 2.3.

In the compact case, a sharp disanalogy between the two flows arises. Every compact mean curvature flow disappears in finite time, so mean curvature flow has no compact steadies, or expanders. On the other hand, in Ricci flow, negative curvature allows a compact solution to last forever, even grow. Therefore, compact steadies and expanders exist, but on sufferance: they are always Einstein [13, Section 2]; [8, Proposition 5.20].

On the other hand, shrinkers typically have finite energy whether compact, or non-compact. Furthermore, the soliton potential need not be constant. (So a Ricci shrinker g(t) need not be Einstein, and a mean curvature shrinker Mt need not be a minimal submanifold of a sphere.) There is accordingly a lively variational calculus for them (see [15, 17, 19, 30, 4] and others).

lit is conjugate to the backward heat equation u t = --Au.

2There are graphs over ~n (due to Ecker) that evolve by mean curvature flow forever, but are asymptotic neither to expanders nor translators. There also appear to be non-compact curves in R 2 whose mean curvature flow exhibits non trivial periodicity, i.e., breathers [3, 19].

Entropy and Reduced Distance for Ricci Expanders

51

1. Expander entropy formula

Let

(M, g(t))

be a solution to the Ricci flow

Ogij

: - 2 R i j 9

Ot

(1.1)

We call

g(t) a (gradient) expander,

if

g(t ) = a(t )fb~ (go) ,

where 4~t is a family of diffeomorphisms whose velocity is given by a gradient vector field, and

a(t)

is any increasing function. Then

a(t) = t - T

for some T and g solves the PDE

R c + V 2 F + g ~ - - O ,

t > T , 2(t - T)

on each time slice, for some function F = F(., t).

Now let M be closed, g a metric, u > 0 a function, and ~r > 0. Dual to the shrinker entropy W [25, Section 3], (which we call W _ ) define the expander entropy by

W + ( g , u , ~ ) : = f M [ ~ ( l V f f , 2

+ R u

)

+ u l o g u

I n

d v + ~ l o g ( 4 ~ r ~ ) + n ,

(1.2/

= fM[~r(lVf+12 + R)-- f+ + n ] u d v

restricted to u satisfying

Here f + = f + ' ~ is defined via

fM

UdV = 1 .

e - f +

b / - -

(4zrcr)n/2 "

Now take u =

u(x, t)

to be a positive solution to the conjugate heat equation

Ou

- - = - - A u + R u ,

Ot

satisfying the conserved condition

f u dv

= 1. Note that f + =

f+,t-T

solves

(1.3)

Of+ _

Af+-q-lVf+[

2 - R n

Ot

2(t - T)

The following monotonicity formula for 142+ was found by varying the signs in the shrinker entropy formula [25, Section 3]. A more elegant derivation is given a few pages hence.

Theorem 1.1 (Expander entropy monotonicity).

Under evolution Equations

(1.1), (1.3)

we have

eij

T) 2

f ~

gij

dv

V~+(g(t), u(t),

t - T) = 2(t -

T)u

+

V i V j f +

-4-

2(t ---- " (1.4)

52 Michael Feldman, Tom llmanen, and Lei Ni

Formula (1.4) is a consequence o f the following differential inequality, which is similar to the one in [25, Section 9] (see also [20]), but with sign changes.

T h e o r e m 1.2.

Then

Let (M, g(t)), u, f+, and T be as above. Define v+ := [(t - T) ( 2 A f + - lV f+12 + R ) - f+ + n ] u ,

0 ) Rij gij 2

+ A - R v+ = 2(t - T)u + V i V j f + + 2(t - T~ '

t > T .

t > T . (1.5)

Proof. We may assume T = 0 . S e t V : = 2 A f + - IVf+12 + R and compute

+ A V = 2[Rij + V i V j f + l 2 + 2 ( V V , V f + ) (1.6)

and

+ A ( t V - f + + n ) = 2 t l R i j + V i V j f + 1 2 + 2 ( V ( t V - f + + n ) , V f + ) + 2 A f + + 2 R + 2 - ~

and do further calculation. [ ]

P r o o f o f T h e o r e m 1.1.

integrate (1.5) over M.

Note that

f

v+ dv = W + . To get the time derivative of l/V+,

[]

R e m a r k . For the steady case, one can define e - f = u, v0 : = Vu = (2 A f - IV f ] 2 + R)u and verify

-~ + A - R vo = 2u tRij + V i V j f l 2 ,

which m a y be viewed as a "neutral" differential Harnack inequality. It may be integrated to get the monotonicity o f .T in [25, Section 1]. See [7].

We next consider the implications for long term behavior of the Ricci flow. Let M be com- pact and consider a Ricci flow g(t) that exists for all t > b. We construct a global solution o f the conjugate heat equation. Let t i ~ 00, and let u i, b < t <_ t i be the solution o f (1.3) with final data

1

bli(x,t i ) " " -- X C M .

v ( t i ) '

Passing t i --+ oo we obtain an immortal solution u > 0 o f (1.3) with fM u = 1 for all t > b. Now let Otj ~ OO and define the blowdown sequence

gj(t) : = otjl g(otjt), t > b/otj .

The following theorem is classic. We give a new proof using l/V+. The hypothesis will hold (subsequentially) if, for example, [Rm [, diameter, and injectivity radius o f g ( t ) / t are controlled for all time [12]. A similar theorem has been proven for the shrinking case [27].

T h e o r e m 1.3 ( H a m i l t o n [13]). Let M be compact. Suppose there are diffeomorphisms dpj such that ~)~(gj(t)) -+ go~(t) smoothly, t > O. Then ge~(t) is an expanding soliton (hence negative Einstein).

Entropy and Reduced Distance for Ricci Expanders 53

Proof. Let u j ( t ) : = ~t;n/2u(otjt). Evidently we can extract a smoothly convergent subse- quence cky(uj) ~ u ~ > 0 satisfying (1.3) for all time. The convergence is smooth, the manifold is compact, and W + is monotone and scale invariant. So 142+ is constant on the limit flow, which

is therefore an expanding soliton. [ ]

R e m a r k . Note that an expanding breather can be expressed as a blowdown of itself, after extending it iteratively to a maximal interval of existence t > b. Applying the theorem then yields an alternative proof that there are no non trivial compact expanding breathers [25, Section 2].

Theorem 1.3 may be proven using any monotone, scale invariant quantity that is constant only on expanders. Hamilton's original method [13] uses the quantity Rmin(/) V(t) 2/n. Altemately (Hamilton), one can use the scaled volume

V(t) :=

V(t)

tn/2 9

Here V(t) is the volume of g(t) and 17'(t) is the volume of the rescaled metric ~(t) : = g ( t ) / t .

Perelman's proof [25, Section 2] uses the scaled Z-functional

~.(g) : = V ( g ) 2 / n Z ( g ) ,

where X(g) : = inf{St'(g,

u)lf

u dv = 1 }. One can also use the forward reduced volume (see Section 2). In each case, the right-hand side of the monotonicity formula vanishes identically only on expanders.

We now drop the hypothesis that the blowdowns converge smoothly to a compact limit. Can we still get convergence to expanders? The various monotonicity formulae will only be useful, if the monotone quantity remains bounded for all time. Accordingly, we embark on a study of the long-term behavior of l/V+, I 7", and ~..

We begin with the scaled volume. Recall from Hamilton [13]

OR

-- A R + 2 l R c l 2 ,

3t

from which follows via R 2 _< n lRc[ 2 and the maximum principle that

n

dV _

l f (

n ) d v < O" (1.7)

R + ~ > O, dt tn/2 R + 2t

So I5"(t) is non increasing, and f ' ~ : = l i m t ~ I5"(t) exists. 3

Next we motivate W + as mentioned above. Analogously to [25, Section 5] (see also [23]), define

A/'(g, u) : = fM u log u dv,

n n

A/'+ (g, u, ~r) : = JV'(g, u) + ~ log(4zr~r) + ~ .

3The proof of Theorem 1.3 (Hamilton) then runs as follows: Assume gj (t) converges smoothly to a compact limit

goo(t). Then (Z(g(t)) ~ ~'(goo(t)), which is therefore constant, so R~ =-- -n/2t, so Rcoo is diagonal and equals -goo/2t. This shows that any such blowdown is negative Einstein. In particular, any compact expander is negative Einstein.

54

Michael Feldman, Tom llmanen, and Lei Ni

The quantity A/" is the Nash entropy [21]. 4 Note that.N'+ is dual to the entropy used for shrinkers in [25, Section 5]. Now let

g(t)

and

u(t)

evolve as above for all t > 0 and set ~r(t) : = t. Compute

dA/" f :__ fM (,vu,2

)

dA/'+ n

dt

+ Ru dr,

dt

- be+ : = be + 2 t ' and

~

(tA/'+) = N'+ + tbe+

f

I

21~

n

n

( IVuul2

) ]

= u l o g u +

+ ~ + ~ + t

+Ru

dv

= ~/V+,

which motivates l/V+.

Next we recall [25, Section 1]

d_f >_ 2 0

dt

n

By O D E comparison, if

g(t)

exists for all t > 0 we have

r/

< be(g, u) < 0,

2 t - t > O .

(Side remark: Passing the starting time to - 0 0 , we notice that a compact eternal solution has be -= 0, hence is a steady soliton. Namely, any compact eternal solution must be a steady soliton. 5)

Next we study A/', IV+, and X as t --+ 00. By Jensen's inequality, we have in general

f u l o g u d v >

l o g ( 1 )

when

f u do

-- 1,

with equality precisely when u = 1 / V . So we have the lower bound

n n

A/'+(g, u, t) > - l o g ( V ( t ) ) + ~ log(4zrt) + ~ .

To get a similar upper bound, we use monotonicity and the construction o f u as asymptotically constant. F r o m above we have

d n

-~A/'+(g, u, t) = •+(g, u, t) = U(g, u) + ~-~ > O,

so A/'+ is monotone non decreasing. This also applies also to u i. So fixing t, we have for all large

t i ,

= + l ~ + 2 "

Taking i --* cx~, combining with the lower bound, and sending t ~ ec, we obtain

n

lim N ' + ( g , u, t) = - l o g (I?~) + ~ (log4~r + 1) .

4Our sign convention for.A/" is the opposite of Nash's.

5 Very recently, a similar result has been proved in [9] for non-compact, two-dimensional eternal solutions, under an assumption on the asymptotic behavior of the conformal factor.

E n t r o p y a n d R e d u c e d D i s t a n c e f o r R i c c i E x p a n d e r s 5 5

(This holds even, if Voo = 0.) If Voo > 0, then since A/'+ is monotone, there exist tj --+ ~ such that

lim t j Y + ( g , u, tj) = O.

j --+ o o

Since W + = t)t-+ + A/'+, and since )42+ is monotone, we obtain

lim )/V+(g, u, t) = lim A/'+(g, u, t) .

t ----~ oo t ----~ oo

(Again, it holds even, if I7'oo = 0.)

Next we turn to k. It is straightforward to show that the minimizer in the definition of k depends smoothly on g, so k ( g ( t ) ) is smooth, and

- - > k 2 , - - < t k < O ,

dt - n 2 -

since g(t) exists for all t > 0. From tj~'+ ~ 0 and the lower bound for k we get

l'l

lim t j k ( g ) = lim tj.T(g, u ) - .

t ) --+ o o t j --+ o o 2

Using k <

f

R dr~ V, (1.7), and the fact that k < 0, we obtain Perelman's monotonicity [25, Section 2],

d -

~ Z ( g ( t ) ) > O. ~21n

Since t - i v ( t ) 2/n ~ ,oo we obtain

lim

~.(g) =

n QL/2

t ----~ o~ - - 2 "

Our results are summarized in the following proposition, which by inspecting the above argument

~

holds even, if V~ = 0.

P r o p o s i t i o n 1.4.

L e t M becompact, g(t) a R i c c i f l o w f o r a l l t > O, a n d u ( t ) thesolution o f the conjugate heat equation constructed above. Then

lim ]/V+(g(t), u(t), t) = - log I)oo + 2 ( 1 + log4zr) ,

t - + o ~

and

lim k ( g ( t ) ) = n ~2/n

t--~ O0 - - 2 v O0 "

R e m a r k . The upper bound on W + can be proven without using A/'+. If ~?oo > 0 then from the monotonicity of t - n / 2 v (t) we get the estimate

fo ~176 fM ( n )

O < t -n/2 R + -~ d v d t < oo .

So there exist t i --+ o0 so that

f n ~

56

Michael Feldman, Tom llmanen, and Lei Ni

Employ this particular sequence t i in the construction of u (it should not affect the value of u, but

we have not proven this) yields for each fixed t and large t i ,

W + ( g ( t ) , u i ( t ) , t ) < ] & + ( g ( t i ) , u i ( t i ) , t i)

, f

-- V(ti ) t i R d v - l ~ 1 7 6 + n --+ - l o g I)oo + 2(1 + log(4rr)), t i --+ oo. The result follows by passing U i - - + U.

As a corollary to Proposition 1.4, we obtain the following estimate.

Proposition

1.5. Let M be compact and let g (t), u (t) solve (1.1), (1.3) for aH t > O. Assume V~ > O. Then the rescaled solution ~([) := g ( t ) / t , ~([) := tn/2u, f+(t) := f+(t), [ = l o g t satisfies

fo~~ fM Rc g 2df) d[ < oo.

(1.8)

This estimate may provide enough control to eventually deduce convergence of a blowdown sequence in a suitable weak sense. We formulate the following natural conjecture as sharply as possible.

Conjecture 1.6 (Long-term behavior).

Let g(t ) be a Ricci flow that exists for all t > 0 on a compact manifold M. Then there is an open set U C M, times tj --~ ~ , and a metric ~ such that

g ( t j ) / t j --+ ~,~ in U ,

where ~ is smooth and negative Einstein, the metric completion of ~,~ has a singular set of dimension at most n - 4, and

v o l =

The metric completion of ~ need not be compact, or connected, but has finite volume, so it should be negative Einstein and not merely an expander. The set U need not be dense in M. If Voo = 0, then take U to be empty and the statement holds vacuously; more generally, the assertion is that the scaled volume of the collapsing parts tends to zero. The rescaling is designed not to see these parts: they recede to infinity, or hide beyond the singular set.

The conjecture is inspired by the convergence picture in the three-dimensional case [ 13, 26]; a good exercise would be to employ the W + monotonicity as an alternative way of recognizing the hyperbolic pieces in that case. The codimension of the singular set is motivated by the fact that the right-hand side f u I Rm + . . 9 ]2 is unitless for n = 4, and is supported by familiar convergence results for Einstein 4-manifolds ([1, 5, 6, 14, 27, 28, 29] and many others), as well as analogous results for mean curvature flow in the critical case [17] and other flows. Later Professor Tian informed us that he had independently formulated a similar conjecture.

Formulation o f # + and v+ functionals. Define, analogously to [25, Section 3], /*+(g, ~r) : = inf )/V+(g, u, ~r), v+(g) := sup/z+(g, or) ,

Entropy and Reduced Distance for Ricci Expanders 57 where u varies over functions satisfying f M u d v = 1. We investigate the existence, smoothness, and monotonicity o f these quantities.

T h e o r e m 1.7. L e t M be a closed manifold.

(a) The inf in the definition o f # + is attained b y a unique u. g + ( g ( t ) , t - T ) is monotone

non decreasing under the Ricci flow, and is constant only on an expander with starting time t = T.

(b) f i X ( g ) < O, then the sup & the definition o f v+ is attained b y a unique ~ . v + ( g ( t ) ) is

monotone non decreasing under the Ricci flow, and is constant only on an expander.

P r o o f (a) Write

-+--

/ [~

+..'I

+.',o..']..

+..

where W 2 = U. This functional is lower semicontinous and coercive o n W 1'2, so it possesses a minimizer w subject to f w 2 ---- 1. Without loss o f generality, w is non negative. By the strong m a x i m u m principle (taking into account the sign o f the nonlinearity), w > 0 and therefore w is smooth. The corresponding u = w 2 minimizes the strictly convex functional

f [.

(4i..l,.i'

+..)

+.,o,.}

,.

+

on the cone o f smooth functions with u > 0, f u = 1, and therefore u is unique. It is straightfor- ward to verify that u varies smoothly when (g, a ) varies smoothly. Write Ug,~ for this minimizer. Now let g(t) be a Ricci flow, fix T 6 R, and let fi(t) : = Ug(t).t-r be the minimizer for each t. Fix to > T and let u(t) t < to be the solution o f the conjugate heat Equation (1.3) with final value if(t0). Since the first variation of W + at t = to with respect to u vanishes, we can compute (smoothly) at t = to d d - ~ # + ( g ( t ) , t - T ) = - ~ W + ( g ( t ) , fi(t), t - T ) = d ~ W + ( g ( t ) , u(t), t - T ) , SO d f R c g 2 - - ~ # + ( g ( t ) , t - T) = ~ u + v Z f + + 2(T - t----~ d v ,

where u realizes the m i n i m u m at t = to. Since to was arbitrary, this formula actually holds for each t, where u realizes the m i n i m u m at time t. If ~ + ( g ( t ) , t - T ) is constant on any interval, the right-hand vanishes, so g(t) is an expander with starting time T.

(b) By the above, # + ( g , c~) is continuous in a , a > 0. We show that it goes to - e ~ at the endpoints. Recall [25, Section 1]

] V u l

5C(g, u) : = + Ru d r , ~.(g) : = inf f ( g , w ) ,

/A u

where u ranges over functions with

f u

= 1. Fix u and estimate

f

"

# + ( g , cr) < W + ( g , u , ~ ) = ~ 5 ( g , u ) + u l o g u d v + ~ l o g ( 4 r r ~ ) + n .

Thus, l~+(g, ~r) --+ - o c as ty --+ 0. I f Z ( g ) < 0, we select u so that 5C(g, u) = Z ( u ) a n d find that

tx+(g, ~ ) ~ - c ~ as cr --+ co. In addition it is strictly concave in ty. Therefore the supremum in

5 8 Michael Feldman, Tom llmanen, and Lei Ni

Now let

g(t)

be a Ricci flow and fix to > O. Since the first variation of/z+(g(t0), or) is zero at cr =

~g(to),

we may compute (smoothly) at t = to

d d d

~v+(g(tl) = -~#+(g(t),

O'g(t)) =

-~Iz+(g(t), cr + t - to).

Thus,

f

24 2

d

R c V2f+ +

dv

-d~v+(g(t)) =

u

q-

where (u, ~r) realizes the minimax at t = to. Since to was arbitrary, this formula actually holds for each t, where (u, or) realizes the minimax at time t. If

v+(g(t))

is ever constant, the right-hand

vanishes, so

g(t)

is an expander. [ ]

2. Forward reduced distance and reduced volume

In this section we derive the dual version of Perelman's monotonicity on the reduced volume in [25, Section 7].

Let

g(t)

solve the Ricci flow on M x [0, T]. Fix x0 and let V be a path (x(0), 0) joining (x0, 0) to (y, t). Analogous to [25], we define

/0'

c+(y) =

~ ( R + I/(0)l 2) d0

(2.1)

Let

X -- Vt(t)

and let Y be a variational vector field along V. One calculates the first variation of s to be

8 ~ + 2 ~ / ~ ( X , y ) : f o t

=

+

~

(

Y, V R - 2 V x X + 4 R c ( X , . ) - ~ X

1 )

do.

(2.2)

From this one can write the s equation:

1 1

V x X - - V R + - - X - 2Rc(X,-)

= 0 . (2.3)

2

2t

This differs from [25] only by the sign in front of

Rc(X, .).

Let

L+(y, t)

denote the length of a shortest s joining (x0, 0) to (y, t). From (2.2) we know that

VL+ = 2 ~ / t X ( t ) . (2.4)

Following closely the computation of [25, Section 7] we obtain

IV L+I 2 = - 4 t R + 4t (R + [XI 2) (2.5)

3L+

- - + ixl2 ( 2 . 6 )

3t

k /

A calculation using the geodesic equation shows that

_

₊

l

_ -

dt

where

H(X)

:=

OR/Ot +

2(VR,

X) + 2Rc(X, X) + R / t

is the exactly twice the traced Li-Yau- Hamilton differential Harnack expression in [11]. (Notice that the H in [25, Section 7] also equals the LYH expression, but evaluated at a negative time t = - r . ) This gives

1

Entropy and Reduced Distance for Ricci Expanders

59

where We then have

fo

t K :=

tI3/2H(X) do 9

2 4 IVL+I 2 =

- 4 t R + - ~ L + + ---~K,

(2.8) OL+ = 2v"tR - 1 K _ _1 L + . (2.9)

Ot

t

2t

These differ from [25] only by the sign in front of K (but recall that the interpretation of H also differs).

We can similarly do the second variation computation. For the a variation vector Y satisfying Y (0) = 0, we get

Jo'(

a2z:+ = 2vq(vrY, x) +

,/~ y . r . R - 2(vry, VxX)

+ 4Y(Rc(Y, X)) - 2X(Rc(Y, Y)) - 2R(X, Y, X, Y)

'

)

+ 2IVxYI 2 - - ( V y Y , X) dr/. /7

(2.10)

Using the s equation we get

fo' (

32/2+ = 2v'7(VyY, X) + v ~

V2R( Y,

Y ) -

2R(X, Y, X, Y)

+ 2]VxY] 2 +

4VyRc(Y, X) - 2VxRc(Y, Y)) drl.

(2.11)

We choose Y such that

1

V x Y = Rc(Y, .) + 7--Y ,

2t

(2.12)

which in particular implies that IY(r/)12 = 0 7]Y(t) 12. Then we obtain

f0 t

1 2 x / ~ ( V 2 R ( y ,

Y)

2R(X, Y, X, Y) +

21Rc(., Y)I 2

V2L+(Y, Y) _<

~ l Y ( t ) l +

2 )

+ -Rc(Y, Y)(4VyRc(Y, X) - 4VxRc(Y, Y)) + 2VxRc(Y, Y) drl.

(2.13)

Now we use the fact that

d (Rc(Y, Y ) ) = ORc Y

1

dt

-Y[-( ' Y) + Vx Rc(Y, Y) +

2IRe(., Y)I 2 +

tgc(V, Y)

which implies

f0 t

2

~/-~VxRc(Y, Y)do = 2~/TRc(Y, Y)

_ f o t ~ (

Rc(Y,Y)3

~

2 OtOec(y,Y)-4lRc(Y,.)12)d~l.

(2.14)

60

Michael Feldman, Tom llmanen, and Lei Ni

Plugging (2.14) into (2.13) we have

[y[2

f0 t

V2L+(Y,

Y) < - - ~ + 2v/tRc(Y, Y) -- ~ H ( X , Y) d o .

Here

H (X, Y)

:= - V 2 R ( y ,

Y) + 2R(X, Y, X, Y) +

2[Rc(., y)[2

1 ORc + - R c ( Y , Y) +

_t

_{2 ~ - ( Y ,}

Y) - 4V~,Rc(Y, X) + 4 V x R c ( Y , Y ) .

This is exactly twice Hamilton's matrix LYH expression. We get the following.

(2.15)

(2.16)

Corollary 2.1.

I f ( M , g(t ) ) has bounded non-negative curvature operator, then

Irl 2

V2L+(Y'

Y) < 7 + 2 v ~ R c ( Y , r ) .

In the general curvature case, tracing (2.15) in Y yields

Now defining

we get

These then imply

n 1

AL+ < ~ + 2 v ~ R - - K . t

1

s

t) := 7 ~ L + ( y , t),

2 4 t

/.+(y, t) := 4ts

t ) ,

s K

17s

: - R + --t + t 3/~ '

0s

K

s

- - R Ot

2t 3/2

t

n

K

As _< R +

2t

2t3/2 "

3s n

- - + A s 1 6 3

z - R - -

< 0 ,

Ot 2t - - A (L+ + 2nt) >_ O,

(2.17)

(2.8')

(2.9')

(2.17')

(2.18)

(2.19)

and negativity of a quantity resembling - v + :

(2As + 17s

- R) - s - n < 0 .

(2.20)

t

Note that s resembles - f + , in contrast to the shrinker case, where s is like f [25, Section 9],

[4]. We have the following theorem from (2.18), (2.19).

Theorem 2.2.

Let g(t) solve the Ricci flow on

M x [0, T].

Then ef+(x,t)

fi(x, t) .--

Entropy and Reduced Distance for Ricci Expanders 61

is a super-solution to the conjugate heat equation, that is. Oft /Ot + Aft - R~ <_ O. In particular, when M is compact, 6 the forward reduced volume

O+(t) = 0~r~176 : = f M fi d v (2.21)

is monotone non increasing along the flow. Furthermore, L+ + 2nt is a super-solution to the standard heat equation.

Remarkably, the entropies I/V and IV+ both increase in forward time, whereas the reduced volumes increase as t approaches the reference time t = 0.

The forward reduced volume is a constant precisely when g (t) is a compact expanding soliton and (x0, 0) is the "vertex" of the soliton. In this case, we get

log O+(t) = - v + ( g ) ,

a constant independent of time. Details will be given in future work; see [4] for the shrinker case. The following proposition says that 0+ is bounded away from zero when r~oo > 0, in the same way as the other monotone quantities.

Proposition 2.3.

The forward reduced volume is bounded below by

logO+(t) > log l?(t) - n _{_ 7(1 + l o g 4 z r ) .}

P r o o f . Since R > - n / 2 t , we get for any curve y from (xo, 0) to (p, t),

f0'

s > ~ - do = - n ~ c q ,

so g+(p, t) >_ - n / 2 and

V (t)e-n/2 (l (t)

O+(t) >_ (4zrt)n/2 (4zre)n/2 . []

This also shows that O+(t) ---> oo as t ~ 0 unless V(t) ~ t n/2 for small t (maximal volume decay). In particular, when the initial metric is smooth except for a conic singularity x0, as may arise by evolution from a smooth metric defined for negative times [10], then O+(t) will not be useful in proving that a blowup about the vertex (x0, 0) for small t > 0 yields an expander.

[1] [2]

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6A localized reduced volume, as well as a localized monotonicity formula, has been found in [24], which removes this restriction on M being compact.

62 Michael Feldman, Tom llmanen, and Lei Ni

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Received December 6, 2004

Department of Mathematics, University of Wisconsin, Madison WI 53706 Departement Mathematik, ETH Zentrum, 8092 Z0rich, Switzerland Department of Mathematics, University of California, San Diego, La Jolla CA 92093

e-maih [email protected]